# Properties

 Degree $2$ Conductor $16$ Sign $0.0846 - 0.996i$ Motivic weight $4$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (2.47 + 3.14i)2-s + (−7.86 + 7.86i)3-s + (−3.78 + 15.5i)4-s + (27.2 − 27.2i)5-s + (−44.1 − 5.30i)6-s + 50.3·7-s + (−58.2 + 26.5i)8-s − 42.8i·9-s + (152. + 18.3i)10-s + (−53.1 − 53.1i)11-s + (−92.5 − 152. i)12-s + (−125. − 125. i)13-s + (124. + 158. i)14-s + 428. i·15-s + (−227. − 117. i)16-s + 286.·17-s + ⋯
 L(s)  = 1 + (0.617 + 0.786i)2-s + (−0.874 + 0.874i)3-s + (−0.236 + 0.971i)4-s + (1.08 − 1.08i)5-s + (−1.22 − 0.147i)6-s + 1.02·7-s + (−0.910 + 0.414i)8-s − 0.528i·9-s + (1.52 + 0.183i)10-s + (−0.438 − 0.438i)11-s + (−0.642 − 1.05i)12-s + (−0.740 − 0.740i)13-s + (0.634 + 0.807i)14-s + 1.90i·15-s + (−0.888 − 0.459i)16-s + 0.990·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0846 - 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0846 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$16$$    =    $$2^{4}$$ Sign: $0.0846 - 0.996i$ Motivic weight: $$4$$ Character: $\chi_{16} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 16,\ (\ :2),\ 0.0846 - 0.996i)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.02804 + 0.944381i$$ $$L(\frac12)$$ $$\approx$$ $$1.02804 + 0.944381i$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2.47 - 3.14i)T$$
good3 $$1 + (7.86 - 7.86i)T - 81iT^{2}$$
5 $$1 + (-27.2 + 27.2i)T - 625iT^{2}$$
7 $$1 - 50.3T + 2.40e3T^{2}$$
11 $$1 + (53.1 + 53.1i)T + 1.46e4iT^{2}$$
13 $$1 + (125. + 125. i)T + 2.85e4iT^{2}$$
17 $$1 - 286.T + 8.35e4T^{2}$$
19 $$1 + (99.5 - 99.5i)T - 1.30e5iT^{2}$$
23 $$1 + 100.T + 2.79e5T^{2}$$
29 $$1 + (-343. - 343. i)T + 7.07e5iT^{2}$$
31 $$1 + 208. iT - 9.23e5T^{2}$$
37 $$1 + (1.15e3 - 1.15e3i)T - 1.87e6iT^{2}$$
41 $$1 - 2.33e3iT - 2.82e6T^{2}$$
43 $$1 + (2.07e3 + 2.07e3i)T + 3.41e6iT^{2}$$
47 $$1 - 1.05e3iT - 4.87e6T^{2}$$
53 $$1 + (-2.13e3 + 2.13e3i)T - 7.89e6iT^{2}$$
59 $$1 + (-3.72e3 - 3.72e3i)T + 1.21e7iT^{2}$$
61 $$1 + (-2.49e3 - 2.49e3i)T + 1.38e7iT^{2}$$
67 $$1 + (329. - 329. i)T - 2.01e7iT^{2}$$
71 $$1 + 1.04e3T + 2.54e7T^{2}$$
73 $$1 + 2.67e3iT - 2.83e7T^{2}$$
79 $$1 + 4.47e3iT - 3.89e7T^{2}$$
83 $$1 + (-1.45e3 + 1.45e3i)T - 4.74e7iT^{2}$$
89 $$1 - 1.14e3iT - 6.27e7T^{2}$$
97 $$1 + 1.31e4T + 8.85e7T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−17.85821045957825609090904445994, −17.03298529416628319144713373686, −16.30213826808910014846143893822, −14.80503495352825476353317421586, −13.38362949683296981123464742261, −11.96364227870492534626312877646, −10.11164253760712221901792607020, −8.276255223790921748716519693415, −5.57777607942541232102608291017, −4.92363426132975832252453235095, 1.99332759993399153954160636058, 5.40183618026703004892783855480, 6.87198751507746545371523975885, 10.01300273275767311285942412036, 11.24862285803189476511894331634, 12.39033133761968601624984824071, 13.88270684440660163084731792241, 14.76522579927562464605458801098, 17.41787396203579471655084913685, 18.13331388844150196708837995523